Irish Leaving Certificate Physics

Fundamental and derived units

Every physical quantity is a number multiplied by a unit, and a number with no unit tells you nothing. Physics builds its units from a small set of fundamental (base) units in the International System, known as SI. The ones used in this course are the metre (m) for length, the kilogram (kg) for mass, the second (s) for time, the ampere (A) for electric current and the kelvin (K) for temperature. The mole and the candela complete the set of seven but rarely appear here.

All other units are derived: they are combinations of the fundamental ones, found by following the definition of the quantity. Speed is distance over time, so its unit is $\mathrm{m\,s^{-1}}$. Acceleration is change of speed over time, so its unit is $\mathrm{m\,s^{-2}}$. Force is mass times acceleration, so one newton is $1\,\mathrm{kg\,m\,s^{-2}}$. Several derived units have their own names because the combination is used so often.

Temperature is usually measured in degrees Celsius in the laboratory, but the SI unit is the kelvin, and a change of one degree Celsius is the same size as a change of one kelvin.

Prefixes and powers of ten

Very large and very small quantities are written with a prefix, or in standard form as a number between 1 and 10 multiplied by a power of ten.

Converting between units means multiplying by a factor equal to one. To change a speed of $72\,\mathrm{km\,h^{-1}}$ into metres per second, write

Take care with units raised to a power, because a conversion factor is raised to the same power. There are $100^3 = 10^6$ cubic centimetres in a cubic metre, not 100, so a density of $1000\,\mathrm{kg\,m^{-3}}$ is $1000\,\mathrm{kg}$ in $10^{6}\,\mathrm{cm^3}$, which is $1\,\mathrm{g\,cm^{-3}}$.

Equations in this course are written for SI units. If you substitute a mass in grams or a length in centimetres without converting, the answer comes out in the wrong unit, and the error is easy to miss.

Significant figures

The number of significant figures in an answer tells the reader how precisely it is known. Leading zeros are not significant, so $0.00420$ has three significant figures, and trailing zeros after a decimal point are. A number such as 1500 is ambiguous, and writing it as $1.5\times10^{3}$ or $1.50\times10^{3}$ settles which is meant.

In a calculation, carry extra figures through the working and round only at the end. The answer is given to the same number of significant figures as the least precise quantity that went into it. If a length of $2.35\,\mathrm{m}$ and a time of $1.2\,\mathrm{s}$ give a speed, the result is $2.35/1.2 = 1.958\ldots$, which is reported as $2.0\,\mathrm{m\,s^{-1}}$, because the time has only two significant figures. Giving $1.958333\,\mathrm{m\,s^{-1}}$ would claim a precision that the measurements never had.

Quantities that are counted or defined exactly, such as the 2 in $2\pi r$ or the number of oscillations in a timing, do not limit the number of significant figures.

Order of magnitude and estimation

The order of magnitude of a quantity is its value rounded to the nearest power of ten. The distance to the Moon, about $3.8\times10^{8}\,\mathrm{m}$, is of order $10^{9}\,\mathrm{m}$, and the diameter of an atom, about $10^{-10}\,\mathrm{m}$, is of order $10^{-10}\,\mathrm{m}$. Comparing orders of magnitude shows how different two scales are: a nucleus, at about $10^{-15}\,\mathrm{m}$, is roughly a hundred thousand times smaller than the atom around it.

An estimate uses round numbers and simple reasoning to find an answer that is correct to within a factor of ten or so. It is useful for checking that a calculated answer is sensible before anything else is done with it.

Worked example: estimate the mass of the air in a classroom. A room about $8\,\mathrm{m}$ by $6\,\mathrm{m}$ with a ceiling $3\,\mathrm{m}$ high has a volume of about $8\times6\times3 = 144\,\mathrm{m^3}$, which is of order $10^{2}\,\mathrm{m^3}$. Air has a density of about $1.2\,\mathrm{kg\,m^{-3}}$, so the mass is about $144\times1.2 \approx 170\,\mathrm{kg}$, of order $10^{2}\,\mathrm{kg}$. A result of $0.2\,\mathrm{kg}$ or of $2\times10^{5}\,\mathrm{kg}$ would be clearly wrong.

Unit analysis

Both sides of a correct physical equation have the same dimensions, meaning the same combination of base units. Checking this is called dimensional or unit analysis, and it catches a wrong formula or a slip in rearranging one. It cannot find a missing dimensionless factor such as $2\pi$ or $\tfrac12$, so a formula can pass the check and still be wrong by that factor.

To test the pendulum formula $T = 2\pi\sqrt{l/g}$, look at the units of the expression under the root. A length divided by an acceleration is

and the square root is a second, which is the unit of a period. The formula passes.

The same method shows that $\tfrac12 m v^2$ is an energy. Mass times speed squared has units $\mathrm{kg\,m^2\,s^{-2}}$, which is a joule. Kinetic energy and work therefore have the same unit, which is why one can be converted into the other.

Units also give a quick test of a final answer. If the question asks for a time and your working ends in $\mathrm{m\,s^{-1}}$, something was inverted along the way.